A210449 Numbers that are the sum of three triangular numbers an odd number of ways.

0, 1, 2, 5, 7, 8, 9, 10, 12, 13, 16, 17, 18, 20, 21, 22, 26, 28, 30, 31, 34, 35, 38, 41, 43, 45, 47, 48, 52, 55, 58, 59, 61, 62, 63, 65, 66, 67, 68, 70, 71, 73, 75, 77, 80, 82, 85, 86, 92, 93, 98, 101, 103, 107, 108, 110, 111, 113, 116, 118, 120, 121, 127

Offset: 1

Author

Alexander Riasanovsky, Jan 20 2013

Keywords

• nonn

Comments

Reduce the elements of A192717 by subtracting 3 and dividing by 8. This makes sense since the elements of A192717 are congruent to 3 (mod 8).

A positive integer n belongs to this sequence precisely when n can be written as t + 2u for triangular numbers t, u an odd number of times, equivalently, written as t + u + v for triangular numbers t, u, v, an odd number of times.

Examples

			For n = 0, 1 representation: 0 + 0 + 0; so 0 belongs to this sequence.
For n = 1, 3 representations: 1 + 0 + 0, 0 + 1 + 0, 0 + 0 + 1; so 1 belongs.
For n = 2, 3 representations: 1 + 1 + 0, 1 + 0 + 1, 0 + 1 + 1; so 2 belongs.
For n = 3, 4 representations: 3 + 0 + 0, 0 + 3 + 0, 0 + 0 + 3, 1 + 1 + 1; so 3 does not belong.
For n = 4, 6 representations: 3 + 1 + 0, 3 + 0 + 1, 1 + 3 + 0, 1 + 0 + 3, 0 + 3 + 1, 0 + 1 + 3; so 4 does not belong.
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Links

• J. N. Cooper, D. Eichhorn, and K. O'Bryant, Reciprocals of binary power series, arXiv:math/0506496 [math.NT], 2005.
• J. N. Cooper, D. Eichhorn, and K. O'Bryant, Reciprocals of binary power series, International Journal of Number Theory, 2 no. 4 (2006), 499-522.
• J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012.

Crossrefs

Cf. A192717, A192628.

Programs

Sage

def BPS(n): #binary power series
    return sum([q^s for s in n])
prec = 2^14
R = PowerSeriesRing(GF(2), 'q', default_prec = prec)
q = R.gen()
tList = [(n*(n+1))//2 for n in range(0, floor(-1+sqrt(8*prec+1))//2)]
tSeries = BPS(tList)
print((tSeries^3).exponents()[:128])